Amid the busy tourist season,
Antonis Tartaras
’ eyewear boutiques in Athens and Corfu are down to about a week’s worth of stock.

He owes foreign suppliers about 20,000 euros ($22,200) for his last delivery of high-end sunglasses. With capital controls making it impossible to pay, some are now asking for upfront payments before they send any more in.

“If the banks don’t open next week or in the next 10 days,” he says. “It will be difficult to continue operations.”

Capital controls in Greece don’t appear yet to have significantly curbed the country’s imports—amounting to about $64 billion a year—especially that of essential items like food, drugs and fuel. That is thanks to safety valves long in place amid years of economic strain here. But for nonessential goods, in particular luxury items, Greek importers are starting to feel the squeeze.

Amid high unemployment and long lines at ATMs, the plight of these mostly small merchants might seem inconsequential in the grand scheme of things. But their new difficulty in sourcing goods could be an early warning of challenges to come for other, more crucial importers if controls aren’t lifted soon.

Over years of Greece’s economic crisis, key exporters and their customers have become heavily reliant on banks and other institutions, including international agencies and insurers, to guarantee payments.

Many Greek importers have obtained letters of credit from their local banks, which then seek a foreign bank guaranteeessentially agreeing to pay the foreign exporter if the Greek customer and the Greek bank are unable to.

International banks, however, had already been asking for cash collateral for such guarantees, according to Michael Spiegel, head of trade finance at
Deutsche Bank AG
. Amid capital controls, that cash can only be transferred from a Greek bank to its foreign counterpart if it gets an exemption from the government. In similar cases in emerging markets, governments only give exemptions for basic goods—fuel, food and medicine, Mr. Spiegel said.

Executives at some big European consumer-goods manufacturers have said so far their exports to Greece have been unaffected. Swiss food giant
Nestlé SA
said it would continue to supply Greece.

“In the current difficult situation, we continue to focus on servicing our consumers and customers,” a Nestlé spokeswoman said, adding: “We are monitoring developments carefully and will adapt as necessary.”

Greece stands on the precipice of exiting the euro despite months of negotiations over a bailout program to manage its debt crisis. Dipti Kapadia shows how the drawn-out talks between the creditors and ruling Syriza party have led to a referendum on Greece’s future. Photo: Getty Images

Another option for exporters to protect against nonpayment: private-sector insurance. In Greece’s case, that insurance is usually supplied by one of three European companies,
Euler Hermes
Group, Atradius NV and Compagnie Française d’Assurance pour le Commerce Extérieur, or
Coface.
Euler Hermes and Atradius both said on Friday that they were awaiting the outcome of a Greek bailout referendum, slated for Sunday, to decide how to proceed. Coface declined to comment specifically about whether it would scale back Greek-related insurance.

Many exporters are also backed by agencies in their home country that provide state-backed export protections—often with the aim of boosting their domestic companies’ ability to compete in risky markets. Public export agencies in Italy, Luxembourg, the U.K. and France all currently supply guarantees for exports to Greece. So far, there is no sign they are cutting off that protection.

“There’s credit available,” said C. Scott M. Shepherd, founder of Northstar Trade Finance Inc., which uses public guarantees to supply financing to exporters, but he said companies aren’t prepared to use it.

Other big suppliers of foreign goods have just gotten use to not being paid. Many large pharmaceutical companies haven’t been paid by Greek buyers for at least six months. They have reassured Greek patients they will nonetheless continue to supply medicine to the country.

Greece’s unpaid drugs bill stood at €1.1 billion in May, according to the most recent estimate by another the European Federation of Pharmaceutical Industries, a trade body.

Goodwill—and lower demand—is also helping keep Greece’s gasoline flowing.
Hellenic Petroleum SA,
35% state owned and a major oil refiner, has relaxed the payment terms for its domestic customers, according to a spokesman. Right now, with domestic demand weak, refiners have been able to afford to pay for their crude suppliers with proceeds from refined product exports.

Things in the luxury-import market aren’t so sanguine. Mr. Tartaras said suppliers, like Italy’s
Luxottica Group
SpA and
Safilo Group
SpA, have been understanding, so far, but are ratcheting up pressure.

“They know what’s going on, and we’ve been doing business for a long time,” he said. “They told us to make payments as soon as we can.” But they’re also asking for upfront payments, instead of the customary predated check.

Luxottica said Greece represents less than 1% of global sales. A spokeswoman said it hasn’t cut off supplies from any clients in Greece, and hasn’t told any of them that they will do so in the near future. They are continuing to send new glasses, even if the clients are behind on payments.

“Our clients are an important asset,” the spokeswoman said. “We are investing in them hoping that things turn around and don’t fall off the cliff.” Safilo said a representative who would be able to comment wasn’t immediately available.

Kostas Kessaris,
a major importer of luxury watches into Greece, says his suppliers are also demanding payment upfront. He says his business doesn’t owe any foreign exporters, “but we can’t get new merchandise” from suppliers.

—Denise Roland, Selina Williams, John Revill and Eric Sylver contributed to this article.